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Asymptotic uniformity of the quantization error for Moran measures on $\mathbb{R}^1$. (arXiv:1802.03723v1 [math.FA])

Let $E$ be a Moran set on $\mathbb{R}^1$ associated with a closed interval
$J$ and two sequences $(n_k)_{k=1}^\infty$ and
$(\mathcal{C}_k=(c_{k,j})_{j=1}^{n_k})_{k\geq1}$. Let $\mu$ be the infinite
product measure (Moran measure) on $E$ associated with a sequence
$(\mathcal{P}_k)_{k\geq1}$ of positive probability vectors with
$\mathcal{P}_k=(p_{k,j})_{j=1}^{n_k},k\geq 1$. We assume that \[
\inf_{k\geq1}\min_{1\leq j\leq n_k}c_{k,j}&gt;0,\;\inf_{k\geq1}\min_{1\leq j\leq
n_k}p_{k,j}&gt;0. \] For every $n\geq 1$, let $\alpha_n$ be an $n$ optimal set in
the quantization for $\mu$ of order $r\in(0,\infty)$ and
$\{P_a(\alpha_n)\}_{a\in\alpha_n}$ an arbitrary Voronoi partition with respect
to $\alpha_n$. For every $a\in\alpha_n$, we write
$I_a(\alpha,\mu):=\int_{P_a(\alpha_n)}d(x,\alpha_n)^rd\mu(x)$ and \[
\underline{J}(\alpha_n,\mu):=\min_{a\in\alpha_n}I_a(\alpha,\mu),\;
\overline{J}(\alpha_n,\mu):=\max_{a\in\alpha_n}I_a(\alpha,\mu). \] We show that
$\underline{J}(\alpha_n,\mu),\overline{J}(查看全文